Bibliography

1: F. V. Atkinson. A divisor problem. The Quarterly Journal of Mathematics, os-12(1):193–200, 1941.
2: R. J. Backlund. Über die Nullstellen der Riemannschen Zetafunktion. Acta Math., 41(1):345–375, 1916.
3: R. C. Baker. The Brun–Titchmarsh theorem. J. Number Theory, 56(2):343–365, 1996.
4: R. C. Baker and G. Harman. The Brun–Titchmarsh theorem on average. In Analytic number theory, Vol. 1 (Allerton Park, IL, 1995), volume 138 of Progr. Math., pages 39–103. Birkhäuser Boston, Boston, MA, 1996.
5: R. C. Baker and G. Harman. The difference between consecutive primes. Proc. London Math. Soc. (3), 72(2):261–280, 1996.
6: R. C. Baker and G. Harman. Shifted primes without large prime factors. Acta Arith., 83(4):331–361, 1998.
7: R. C. Baker, G. Harman, and J. Pintz. The exceptional set for Goldbach’s problem in short intervals. In Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), volume 237 of London Math. Soc. Lecture Note Ser., pages 1–54. Cambridge Univ. Press, Cambridge, 1997.
8: R. C. Baker, G. Harman, and J. Pintz. The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83(3):532–562, 2001.
9: S. Baluyot. On the zeros of Riemann’s zeta-function. PhD thesis, University of Rochester, 2017.
10: W. D. Banks and I. E. Shparlinski. Bounds on short character sums and \(L\)–functions with characters to a powerful modulus. Journal d’Analyse Mathématique, 139:239–263, 2019.
11: D. Bazzanella. Primes in almost all short intervals. II. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 3(3):717–726, 2000.
12: D. Bazzanella and A. Perelli. The exceptional set for the number of primes in short intervals. J. Number Theory, 80(1):109–124, 2000.
13: C. Bellotti. Explicit zero density estimate near unity, 2023.
14: C. Bellotti. An explicit log-free zero density estimate for the Riemann zeta-function, 2025.
15: C. Bellotti and A. Yang. On the generalised Dirichlet divisor problem. Bulletin of the London Mathematical Society, 56(5):1859–1878, May 2024.
16: M. Bennett. Fractional parts of powers of rational numbers. Mathematical Proceedings of the Cambridge Philosophical Society, 114(2):191–201, 1993.
17: E. Bombieri. Le grand crible dans la théorie analytique des nombres, volume No. 18 of Astérisque. Société Mathématique de France, Paris, 1974. Avec une sommaire en anglais.
18: E. Bombieri and H. Iwaniec. On the order of \(\zeta (\frac{1}{2} + it)\). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Ser. 4, 13(3):449–472, 1986.
19: J. Bourgain. Remarks on Montgomery’s conjectures on Dirichlet sums. In Geometric aspects of functional analysis (1989–90), volume 1469 of Lecture Notes in Math., pages 153–165. Springer, Berlin, 1991.
20: J. Bourgain. Remarks on Halász-Montgomery Type Inequalities. In J. Lindenstrauss and V. Milman, editors, Geometric Aspects of Functional Analysis, pages 25–39. Birkhäuser Basel, Basel, 1995.
21: J. Bourgain. On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function. International Mathematics Research Notices, 2000(3):133–146, 2000.
22: J. Bourgain. On the distribution of Dirichlet sums. II. In Number theory for the millennium, I (Urbana, IL, 2000), pages 87–109. A K Peters, Natick, MA, 2002.
23: J. Bourgain. Decoupling, exponential sums and the Riemann zeta function. Journal of the American Mathematical Society, 30(1):205–224, 2017.
24: J. Bourgain, C. Demeter, and L. Guth. Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2), 184(2):633–682, 2016.
25: J. Bourgain and M. Z. Garaev. Kloosterman sums in residue rings. Acta Arith., 164(1):43–64, 2014.
26: J. Brüdern and T. D. Wooley. On Waring’s problem for larger powers, 2022.
27: D. A. Burgess. On character sums and \(L\)-series. Proc. London Math. Soc. (3), 12:193–206, 1962.
28: D. A. Burgess. On character sums and \(L\)-series. II. Proc. London Math. Soc. (3), 13:524–536, 1963.
29: F. Carlson. Über die nullstellen der dirichletschen reihen und der riemannschen \(\zeta \)-funktion. Ark. Mat. Astron. Fys., 15(20):28, 1921.
30: W. Castryck, É. Fouvry, G. Harcos, E. Kowalski, P. Michel, P. Nelson, E. Paldi, J. Pintz, A. Sutherland, T. Tao, and X.-F. Xie. New equidistribution estimates of Zhang type. Algebra & Number Theory, 8(9):2067–2199, December 2014.
31: B. Chen. Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet’s \(L\)-functions, 2025.
32: B. Chen, G. Debruyne, and J. Vindas. On the density hypothesis for \(l\)-functions associated with holomorphic cusp forms. Revista Matemática Iberoamericana, 2024. Published online first.
33: J.-R. Chen. On the divisor problem for \(d_3(n)\). Sci. Sinica, 14:19–29, 1965.
34: J.-R. Chen. On the least prime in an arithmetical progression. Sci. Sinica, 14:1868–1871, 1965.
35: J.-R. Chen. On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet’s \(L\)-functions. Sci. Sinica, 20(5):529–562, 1977.
36: J.-R. Chen. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s \(L\)-functions. II. Sci. Sinica, 22(8):859–889, 1979.
37: J.-R. Chen and J.-M. Liu. On the least prime in an arithmetical progression. III. Sci. China Ser. A, 32(6):654–673, 1989.
38: J.-R. Chen and J.-M. Liu. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s \(L\)-functions. V. In International symposium on number theory and analysis in memory of Hua Loo Keng, held August 1-7, 1988 at the Tsing Hua University, Beijing, China. Volume I: Number theory, pages 19–42. Berlin: Springer-Verlag; Beijing: Science Press, 1991.
39: T.-T. Chih. A divisor problem. Acad. Sinica Sci. Record, 3:177–182, 1950.
40: S. Chourasiya. An explicit version of Carlson’s theorem, 2024. arXiv:2412.02068.
41: S. Chourasiya and A. Simonič. An explicit form of Ingham’s zero density estimate, 2025.
42: L. Cladek and T. Tao. Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle. Ars Inven. Anal., pages Paper No. 1, 38, 2021.
43: J. B. Conrey. At least two fifths of the zeros of the Riemann zeta function are on the critical line. Bulletin (New Series) of the American Mathematical Society, 20(1):79–81, 1989.
44: R. J. Cook. On the occurrence of large gaps between prime numbers. Glasgow Math. J., 20(1):43–48, 1979.
45: J. G. v. d. Corput. Verschärfung der Abschätzung beim Teilerproblem. Mathematische Annalen, 87(1-2):39–65, March 1922.
46: J. G. v. d. Corput. Zum Teilerproblem. Mathematische Annalen, 98(1):697–716, March 1928.
47: H. Cramér. Über das Teilerproblem von Piltz. Arkiv för Mat. Astr. och. Fysik, 16(21), 1922.
48: H. Cramér. On the order of magnitude of the difference between consecutive prime numbers. Acta Arith., 2:23–46, 1936.
49: G. Csordas, T. S. Norfolk, and R. S. Varga. A lower bound for the de Bruijn-Newman constant \(\Lambda \). Numer. Math., 52(5):483–497, 1988.
50: G. Csordas, A. M. Odlyzko, W. Smith, and R. S. Varga. A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant \(\Lambda \). Electron. Trans. Numer. Anal., 1:104–111 (electronic only), 1993.
51: G. Csordas, A. Ruttan, and R. S. Varga. The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis. Numerical Algorithms, 1(2):305–329, 1991.
52: G. Csordas, W. Smith, and R. S. Varga. Lehmer pairs of zeros, the de Bruijn-Newman constant \(\Lambda \), and the Riemann hypothesis. Constr. Approx., 10(1):107–129, 1994.
53: H. Davenport. On waring’s problem for fourth powers. Annals of Mathematics, 40(4):731–747, 1939.
54: H. Davenport. On Waring’s problem for cubes. Acta Mathematica, 71(0):123–143, 1939.
55: N. G. de Bruijn. The roots of trigonometric integrals. Duke Math. J., 17:197–226, 1950.
56: C. J. de La Vallée Poussin. Recherches analytiques sur la théorie des nombres premiers. Annales de la Société scientifique de Bruxelles, 1896.
57: C. J. de La Vallée Poussin. Sur la fonction [zeta](s) de Riemann et le nombre des nombres premiers inferieurs à une limite donnée. Mémoires couronnés et autres mémoires / Collection in-8°. Hayez, 1899.
58: C. Demeter, L. Guth, and H. Wang. Small cap decouplings. Geometric and Functional Analysis, 30(4):989–1062, 08 2020.
59: J. M. Deshouillers and H. Iwaniec. On the Brun–Titchmarsh theorem on average. In Topics in Classical Number Theory, pages 319–333. Math. Soc. János Bolyai 34, Elsevier, North Holland, Amsterdam, 1984.
60: A. Dobner. A proof of Newman’s conjecture for the extended Selberg class. Acta Arith., 201(1):29–62, 2021.
61: A. K. Dubitskas. A lower bound for the quantity \(||(3/2)k||\). Russian Mathematical Surveys, 45:163 – 164, 1990.
62: J. Duttlinger and W. Schwarz. Über die Verteilung der pythagoräischen Dreiecke. Colloq. Math., 43(2):365–372, 1980.
63: J. Elliott. Analytic number theory and algebraic asymptotic analysis. 2024.
64: P. Erdős. On the arithmetical density of the sum of two sequences one of which forms a basis for the integers. Acta Arithmetica, 1(2):197–200, 1935.
65: P. Erdős. On the difference of consecutive primes. The Quarterly Journal of Mathematics, os-6(1):124–128, 1935.
66: E. Fogels. On the zeros of \(L\)-functions. Acta Arith., 11:67–96, 1965.
67: K. Ford. Vinogradov’s integral and bounds for the Riemann zeta function. Proc. London Math. Soc. (3), 85(3):565–633, 2002.
68: K. Ford, B. Green, S. Konyagin, J. Maynard, and T. Tao. Long gaps between primes. Journal of the American Mathematical Society, 31(1):65–105, 2017.
69: K. Ford, B. Green, S. Konyagin, and T. Tao. Large gaps between consecutive prime numbers. Annals of Mathematics, 183(3):935–974, 2016.
70: M. Forti and C. Viola. Density estimates for the zeros of \(L\)-functions. Acta Arith., 23:379–391, 1973.
71: M. Forti and C. Viola. On large sieve type estimates for the Dirichlet series operator. In Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), volume Vol. XXIV of Proc. Sympos. Pure Math., pages 31–49. Amer. Math. Soc., Providence, RI, 1973.
72: É. Fouvry. Sur le théorème de Brun–Titchmarsh. Acta Arith., 43:417–424, 1984.
73: É. Fouvry. Théorème de Brun–Titchmarsh: application au théorème de Fermat. Invent. Math., 79:383–407, 1985.
74: É. Fouvry and M. Radziwiłł. Level of distribution of unbalanced convolutions. Ann. Sci. Éc. Norm. Supér., 55:537–568, 2022.
75: J. B. Friedlander and H. Iwaniec. The Brun-Titchmarsh theorem. In Analytic number theory (Kyoto, 1996), volume 247 of London Math. Soc. Lecture Note Ser., pages 85–93. Cambridge Univ. Press, Cambridge, 1997.
76: J. B. Friedlander and H. Iwaniec. Exceptional characters and prime numbers in arithmetic progressions. Int. Math. Res. Not., (37):2033–2050, 2003.
77: J. B. Friedlander and H. Iwaniec. Exceptional characters and prime numbers in short intervals. Selecta Mathematica, 10:61–69, 2004.
78: A. Gafni and T. Tao. On the number of exceptional intervals to the prime number theorem in short intervals, 2025.
79: P. X. Gallagher. A large sieve density estimate near \(\sigma =1\). Invent. Math., 11:329–339, 1970.
80: P. X. Gallagher and J. H. Mueller. Primes and zeros in short intervals. 1978(303-304):205–220, 1978.
81: D. M. Goldfeld. A further improvement of the Brun-Titchmarsh theorem. J. London Math. Soc. (2), 11(4):434–444, 1975.
82: D. Goldston, J. Pintz, and C. Y. Yıldırım. Primes in tuples I. Annals of Mathematics, 170(2):819–862, September 2009.
83: D. Goldston, J. Pintz, and C. Y. Yıldırım. Primes in tuples II. Acta Mathematica, 204(1):1–47, 2010.
84: S. Graham. Applications of sieve methods. ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)–University of Michigan.
85: S. Graham. An asymptotic estimate related to Selberg’s sieve. J. Number Theory, 10(1):83–94, 1978.
86: S. Graham. On Linnik’s constant. Acta Arith., 39(2):163–179, 1981.
87: E. Grosswald. Sur l’ordre de grandeur des différences \(\psi (x)−x\) et \(\pi (x)−\operatorname {li}(x)\). C.R. Acad. Sci. Paris, 260:3813–3816, 1965.
88: L. Guth and J. Maynard. New large value estimates for dirichlet polynomials, 2024.
89: J. Hadamard. Sur la distribution des zéros de la fonction \(\zeta (s)\) et ses conséquences arithmétiques. Bulletin de la Société Mathématique de France, 24:199–220, 1896.
90: G. Halász. On the average of multiplicative number-theoretic functions. Acta Math. Hungar., 19:365–404, 1968.
91: G. Halász and P. Turán. On the distribution of roots of Riemann zeta and allied functions, I. Journal of Number Theory, 1(1):121–137, 1969.
92: W. Haneke. Verschärfung der abschätzung von \(\zeta (1/2+it)\). Acta Arithmetica, 8(4):357–430, 1963.
93: G. H. Hardy. On Dirichlet’s divisor problem. Proc. London Math. Soc. (2), 15:1–25, 1916.
94: G. H. Hardy. The average order of the arithmetical functions \(p(x)\) and \(\delta (x)\). Proceedings of the London Mathematical Society, s2-15(1):192–213, 1917.
95: G. H. Hardy and J. E. Littlewood. Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Mathematica, 41(0):119–196, 1916.
96: G. H. Hardy and J. E. Littlewood. The approximate functional equation in the theory of the zeta-function, with applications to the divisor-problems of Dirichlet and Piltz. Proceedings of the London Mathematical Society, s2-21(1):39–74, 1923.
97: G. H. Hardy and J. E. Littlewood. On Lindelöf’s hypothesis concerning the Riemann zeta-function. Proc. R. Soc. A, pages 403–412, 1923.
98: G. H. Hardy and J. E. Littlewood. Some problems of partitio numerorum (VI): Further researches in Waring’s problem. Mathematische Zeitschrift, 23(1):1–37, December 1925.
99: G. Harman. Primes in short intervals. Math. Z., 180(3):335–348, 1982.
100: G. Harman. On the distribution of \(\alpha p\) modulo one. J. London Math. Soc. (2), 27(1):9–18, 1983.
101: G. Harman. Prime-detecting sieves, volume 33 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2007.
102: D. R. Heath-Brown. The differences between consecutive primes. J. London Math. Soc. (2), 18(1):7–13, 1978.
103: D. R. Heath-Brown. The twelfth power moment of the Riemann-function. The Quarterly Journal of Mathematics, 29(4):443–462, 1978.
104: D. R. Heath-Brown. The differences between consecutive primes. II. J. London Math. Soc. (2), 19(2):207–220, 1979.
105: D. R. Heath-Brown. The differences between consecutive primes. III. J. London Math. Soc. (2), 20(2):177–178, 1979.
106: D. R. Heath-Brown. A large values estimate for Dirichlet polynomials. Journal of the London Mathematical Society, s2-20(1):8–18, 1979.
107: D. R. Heath-Brown. Zero density estimates for the Riemann zeta-function and Dirichlet L -functions. Journal of the London Mathematical Society, s2-19(2):221–232, 1979.
108: D. R. Heath-Brown. Mean values of the zeta-function and divisor problems. In Recent Progress in Analytic Number Theory, volume 1, pages 115–119, Durham 1979, 1981. Academic, London.
109: D. R. Heath-Brown. Gaps between primes, and the pair correlation of zeros of the zeta-function. Acta Arithmetica, 41(1):85–99, 1982.
110: D. R. Heath-Brown. Finding primes by sieve methods. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 487–492. PWN, Warsaw, 1984.
111: D. R. Heath-Brown. Siegel zeros and the least prime in an arithmetic progression. The Quarterly Journal of Mathematics Oxford, 41(2):405–418, 1990.
112: D. R. Heath-Brown. Zero-free regions for Dirichlet \(L\)-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. (3), 64(2):265–338, 1992.
113: D. R. Heath-Brown. A new \(k\)th derivative estimate for exponential sums via Vinogradov’s mean value. Proceedings of the Steklov Institute of Mathematics, 296(1):88–103, 2017.
114: D. R. Heath-Brown. The differences between consecutive primes, V. International Mathematics Research Notices, 2021(22):17514–17562, 2021.
115: D. R. Heath-Brown and H. Iwaniec. On the difference between consecutive primes. Invent. Math., 55(1):49–69, 1979.
116: H. Heilbronn. Über den Primzahlsatz von Herrn Hoheisel. Math. Z., 36(1):394–423, 1933.
117: H. A. Helfgott. The ternary Goldbach problem, 2015.
118: G. Hoheisel. Primzahlprobleme in der analysis. Siz. Preuss. Akad. Wiss., 33:580–588, 1930.
119: C. Hooley. On the Brun–Titchmarsh theorem. J. Reine Angew. Math., 255:60–79, 1972.
120: C. Hooley. On the largest prime factor of \(p + a\). Mathematika, 20:135–143, 1973.
121: L. K. Hua. The lattice-points in a circle. The Quarterly Journal of Mathematics, os-13(1):18–29, 1942.
122: M. N. Huxley. On the difference between consecutive primes. Invent. Math., 15:164–170, 1972.
123: M. N. Huxley. Large values of Dirichlet polynomials. Acta Arithmetica, 24(4):329–346, 1973.
124: M. N. Huxley. Large values of Dirichlet polynomials II. Acta Arithmetica, 27:159–170, 1975.
125: M. N. Huxley. Large values of Dirichlet polynomials, III. Acta Arithmetica, 26(4):435–444, 1975.
126: M. N. Huxley. A note on large gaps between prime numbers. Acta Arith., 38(1):63–68, 1980/81.
127: M. N. Huxley. Exponential Sums and Lattice Points II. Proceedings of the London Mathematical Society, s3-66(2):279–301, March 1993.
128: M. N. Huxley. Exponential sums and the Riemann zeta function IV. Proceedings of the London Mathematical Society, s3-66(1):1–40, 1993.
129: M. N. Huxley. Area, Lattice Points, and Exponential Sums. Number New Ser., 13 in Oxford Science Publications. Clarendon Press; Oxford University Press, Oxford; New York, 1996.
130: M. N. Huxley. Exponential sums and lattice points III. Proceedings of the London Mathematical Society, 87(03):591–609, 2003.
131: M. N. Huxley. Exponential sums and the Riemann zeta function V. Proceedings of the London Mathematical Society, 90(01):1–41, 2005.
132: M. N. Huxley and G. Kolesnik. Exponential sums and the Riemann zeta function III. Proceedings of the London Mathematical Society, s3-62(3):449–468, 1991.
133: M. N. Huxley and G. Kolesnik. Exponential sums and the Riemann zeta function III. Proceedings of the London Mathematical Society, s3-66(2):302–302, 1993.
134: M. N. Huxley and G. Kolesnik. Exponential sums with a large second derivative. In Matti Jutila and Tauno Metsänkylä, editors, Number Theory, pages 131–144. De Gruyter, Berlin, Boston, 2001.
135: M. N. Huxley and N. Watt. Exponential sums and the Riemann zeta function. Proceedings of the London Mathematical Society, s3-57(1):1–24, 1988.
136: M. N. Huxley and N. Watt. The Hardy-Littlewood method for exponential sums. In Number theory, Vol. I (Budapest, 1987), volume 51 of Colloq. Math. Soc. János Bolyai, pages 173–191. North-Holland, Amsterdam, 1990.
137: A. E. Ingham. On the difference between consecutive primes. The Quarterly Journal of Mathematics, os-8(1):255–266, 1937.
138: A. E. Ingham. On the estimation of \({N}(\sigma , t)\). The Quarterly Journal of Mathematics, os-11(1):201–202, 1940.
139: A. E. Ingham. The distribution of prime numbers. Number no. 30 in Cambridge mathematical library. Cambridge University Press, Cambridge ; New York, 1990.
140: A. Ivić. On sums of large differences between consecutive primes. Math. Ann., 241(1):1–9, 1979.
141: A. Ivić. Exponent pairs and the zeta function of Riemann. Studia Sci. Math. Hungar., 15(1-3):157–181, 1980.
142: A. Ivić. Topics in recent zeta function theory. Publications mathématiques d’Orsay. Université de Paris-Sud, Département de Mathématique, 1983.
143: A. Ivić. A zero-density theorem for the Riemann zeta-function. Tr. Mat. Inst. Steklova, 163:85–89, 1984.
144: A. Ivić. The Riemann zeta-function. Dover Publications, Inc., Mineola, NY, 2003. Theory and applications, Reprint of the 1985 original [Wiley, New York; MR0792089 (87d:11062)].
145: A. Ivić and M. Ouellet. Some new estimates in the dirichlet divisor problem. Acta Arithmetica, 52(3):241–253, 1989.
146: A. Ivić. A note on the zero-density estimates for the zeta function. Archiv der Mathematik, 33(1):155–164, 1979.
147: H. Iwaniec. On the Brun-Titchmarsh theorem. J. Math. Soc. Japan, 34(1):95–123, 1982.
148: H. Iwaniec and M. Jutila. Primes in short intervals. Ark. Mat., 17(1):167–176, 1979.
149: H. Iwaniec and E. Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.
150: H. Iwaniec and C. J. Mozzochi. On the divisor and circle problems. Journal of Number Theory, 29(1):60–93, May 1988.
151: H. Iwaniec and J. Pintz. Primes in short intervals. Monatsh. Math., 98(2):115–143, 1984.
152: O. Järviniemi. On large differences between consecutive primes, 2022.
153: C. Jia. Difference between consecutive primes. Sci. China Ser. A, 38(10):1163–1186, 1995.
154: C. Jia. Goldbach numbers in short interval. I. Sci. China Ser. A, 38(4):385–406, 1995.
155: C. Jia. Almost all short intervals containing prime numbers. Acta Arith., 76(1):21–84, 1996.
156: C. Jia. On the exceptional set of Goldbach numbers in a short interval. Acta Arith., 77(3):207–287, 1996.
157: M. Jutila. A new estimate for Linnik’s constant. Ann. Acad. Sci. Fenn. Ser. A I, 471:8, 1970.
158: M. Jutila. On two theorems of Linnik concerning the zeros of Dirichlet’s L-functions. Ph.D. Thesis, Turun yliopisto, 1970.
159: M. Jutila. On Linnik’s constant. Math. Scand., 41(1):45–62, 1977.
160: M. Jutila. Zero-density estimates for L-functions. Acta Arithmetica, 32(1):55–62, 1977.
161: M. Jutila. Zeros of the zeta-function near the critical line. In Studies in pure mathematics, pages 385–394. Birkhäuser, Basel, 1983.
162: M. Jutila. On a density theorem of H. L. Montgomery for \(L\)-functions. Ann. Acad. Sci. Fenn. Ser. A I, 520:1–13, 1984.
163: H. Kadiri, A. Lumley, and N. Ng. Explicit zero density for the Riemann zeta function. Journal of Mathematical Analysis and Applications, 465(1):22–46, 2018.
164: K. Kawada and T. D. Wooley. On the Waring–Goldbach problem for fourth and fifth powers. Proceedings of the London Mathematical Society, 83(1):1–50, 2001.
165: B. Kerr. Large values of Dirichlet polynomials and zero density estimates for the Riemann zeta function, 2019.
166: H. Ki, Y.-O. Kim, and J. Lee. On the de Bruijn-Newman constant. Adv. Math., 222(1):281–306, 2009.
167: H. H. Kim and P. Sarnak. Refined estimates towards the Ramanujan and Selberg conjectures. J. Amer. Math. Soc, 16(1):175–181, 2003.
168: H. Koch. Sur la distribution des nombres premiers. Acta Mathematica, 24(0):159–182, 1901.
169: G. Kolesnik. The improvement of the remainder term in the divisor problem. Mat. Zametki, 6:545–554, 1969.
170: G. Kolesnik. An estimate for certain trigonometric sums. Acta Arith., 25:7–30. (errata insert), 1973/74.
171: G. Kolesnik. On the estimation of multiple exponential sums. In Recent progress in analytic number theory, Vol. 1 (Durham, 1979), pages 231–246. Academic Press, London-New York, 1981.
172: G. Kolesnik. On the order of \(\zeta (\frac{1}{2}+it)\) and \(\Delta (R)\). Pacific Journal of Mathematics, 98(1):107–122, 1982.
173: G. Kolesnik. On the method of exponent pairs. Acta Arith., 45(2):115–143, 1985.
174: A. V. Kumchev and T. D. Wooley. On the Waring–Goldbach problem for seventh and higher powers. Monatshefte für Mathematik, 183(2):303–310, 2016.
175: J. Lambek and L. Moser. On the distribution of Pythagorean triangles. Pacific J. Math., 5:73–83, 1955.
176: E. Landau. Über die anzahl der gitterpunkte in geweissen bereichen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1912:687–770, 1912.
177: A. V. Lelechenko. Linear programming over exponent pairs. Acta Universitatis Sapientiae, Informatica, 5(2):271–287, 2014.
178: H. Li. Goldbach numbers in short intervals. Sci. China Ser. A, 38(6):641–652, 1995.
179: H. Li. Primes in short intervals. Math. Proc. Cambridge Philos. Soc., 122(2):193–205, 1997.
180: R. Li. Hybrid estimation of single exponential sums, exceptional characters and primes in short intervals, 2024.
181: R. Li. Primes in almost all short intervals, 2024.
182: R. Li. An average Brun–Titchmarsh theorem and shifted primes with a large prime factor, 2025.
183: R. Li. The number of primes in short intervals and numerical calculations for Harman’s sieve, 2025.
184: R. Li. On the generalized Dirichlet divisor problem, 2025.
185: R. Li. Primes in almost all short intervals II, 2025.
186: X. Li and X. Yang. An improvement on Gauss’s circle problem and Dirichlet’s divisor problem, 2023.
187: Y. V. Linnik. On Erdős’s theorem on the addition of numerical sequences. Mat. Sb., Nov. Ser., 10:67–78, 1942.
188: Y. V. Linnik. An elementary solution of the problem of Waring by Schnirelman’s method. Matematiceskij sbornik, 54(2):225–230, 1943.
189: Y. V. Linnik. On the representation of large numbers as sums of seven cubes. Mat. Sb., Nov. Ser., 12:218–224, 1943.
190: Y. V. Linnik. On the least prime in an arithmetic progression. I. The basic theorem. Rec. Math. [Mat. Sbornik] N.S., 15/57:139–178, 1944.
191: Y. V. Linnik. On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S., 15/57:347–368, 1944.
192: J. E. Littlewood. Sur la distribution des nombres premiers. C. R. Acad. Sci., Paris, 158:1869–1872, 1914.
193: J. E. Littlewood. Researches in the theory of the Riemann \(\zeta \)-function. Proc. London Math. Soc., 20(2):22–27, 1922.
194: J. E. Littlewood, A. Walfisz, and E. Landau. The lattice points of a circle. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 106(739):478–488, 1924.
195: J. Liu, T. D. Wooley, and G. Yu. The quadratic Waring–Goldbach problem. Journal of Number Theory, 107(2):298–321, 2004.
196: S. Lou and Q. Yao. Difference between consecutive primes. Chinese Journal of Nature (Nature Journal), 7(9):713, 1984.
197: S. Lou and Q. Yao. On the upper bound of difference between consecutive primes. Kexue Tongbao, 30(8):1127–1128, 1985.
198: S. Lou and Q. Yao. On the Brun-Titchmarsh theorem. Ziran Zazhi, 5(393), 1986.
199: S. Lou and Q. Yao. Upper bounds for primes in intervals. Chinese Ann. Math. Ser. A, 10(3):255–262, 1989.
200: S. Lou and Q. Yao. A Chebychev’s type of prime number theorem in a short interval. II. Hardy-Ramanujan J., 15:1–33, 1992.
201: S. Lou and Q. Yao. The number of primes in a short interval. Hardy-Ramanujan Journal, 16:127, 1993.
202: K. Mahler. On the fractional parts of the powers of a rational number (ii). Mathematika, 4(2):122–124, 1957.
203: H. Maier. Primes in short intervals. Michigan Math. J., 32(2):221–225, 1985.
204: H. Maier and C. Pomerance. Unusually large gaps between consecutive primes. Transactions of the American Mathematical Society, 322(1):201–237, 1990.
205: K. Matomäki. Large differences between consecutive primes. The Quarterly Journal of Mathematics, 58(4):489–518, 2007.
206: K. Matomäki and J. Teräväinen. A note on zero density results implying large value estimates for Dirichlet polynomials, 2024.
207: J. Maynard. On the difference between consecutive primes, 2012.
208: J. Maynard. On the Brun-Titchmarsh theorem. Acta Arith., 157(3):249–296, 2013.
209: J. Maynard. Small gaps between primes. Annals of Mathematics, pages 383–413, 2015.
210: J. Maynard. Large gaps between primes. Annals of Mathematics, 183(3):915–933, 2016.
211: Z. Meng. The distribution of the zeros of \(L\)-functions and the least prime in some arithmetic progression. Sci. China Ser. A, 43(9):937–944, 2000.
212: Z. Meng. A note on the least prime in an arithmetic progression. Acta Scientiarum Naturalium Universitatis Pekinensis, 37(1):20–22, 2001.
213: Z. Meng. A note on the Linnik’s constant, 2010.
214: H. Menzer. On the number of primitive Pythagorean triangles. Math. Nachr., 128:129–133, 1986.
215: H. Mikawa. On the Brun-Titchmarsh theorem. Tsukuba J. Math., 15(1):31–40, 1991.
216: S.-H. Min. On the order of \(\zeta (1/2+it)\). Transactions of the American Mathematical Society, 65(3):448–472, 1949.
217: H. L. Montgomery. Zeros of \(L\)-functions. Invent. Math., 8:346–354, 1969.
218: H. L. Montgomery. Topics in multiplicative number theory, volume Vol. 227 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1971.
219: H. L. Montgomery and R. C. Vaughan. The large sieve. Mathematika, 20:119–134, 1973.
220: Y. Motohashi. On some improvements of the Brun-Titchmarsh theorem. II. Sūrikaisekikenkyūsho Kokyūroku, (193):97–109, 1973.
221: Y. Motohashi. On some improvements of the Brun-Titchmarsh theorem. J. Math. Soc. Japan, 26:306–323, 1974.
222: C. J. Mozzochi. On the difference between consecutive primes. J. Number Theory, 24(2):181–187, 1986.
223: C. M. Newman. Fourier transforms with only real zeros. Proc. Amer. Math. Soc., 61(2):245–251 (1977), 1976.
224: C. M. Newman and W. Wu. Constants of de Bruijn–Newman type in analytic number theory and statistical physics. Bull. Amer. Math. Soc. (N.S.), 57(4):595–614, 2020.
225: L. W. Nieland. Zum Kreisproblem. Mathematische Annalen, 98(1):717–736, March 1928.
226: T. S. Norfolk, A. Ruttan, and R. S. Varga. A lower bound for the de Bruijn-Newman constant \(\Lambda \). II. In Progress in approximation theory (Tampa, FL, 1990), volume 19 of Springer Ser. Comput. Math., pages 403–418. Springer, New York, 1992.
227: A. M. Odlyzko. An improved bound for the de Bruijn-Newman constant. volume 25, pages 293–303. 2000. Mathematical journey through analysis, matrix theory and scientific computation (Kent, OH, 1999).
228: C. D. Pan. On the least prime in an arithmetical progression. Sci. Record (N.S.), 1:311–313, 1957.
229: C. D. Pan. On the least prime in an arithmetical progression. Acta. Sci. Natur. Univ. Pekinensis, 4:1–34, 1958.
230: A. S. Peck. On the differences between consecutive primes. PhD thesis, University of Oxford, 1996.
231: A. S. Peck. Differences between consecutive primes. Proceedings of the London Mathematical Society, 76(1):33–69, 1998.
232: E. Phillips. The zeta-function of Riemann; further developments of van der Corput’s method. The Quarterly Journal of Mathematics, os-4(1):209–225, 1933.
233: J. Pintz. On primes in short intervals. I. Studia Sci. Math. Hungar., 16(3-4):395–414, 1981.
234: J. Pintz. Very large gaps between consecutive primes. Journal of Number Theory, 63(2):286–301, 1997.
235: J. Pintz. Some new results on gaps between consecutive primes. Number Theory, Analysis, and Combinatorics: Proceedings of the Paul Turan Memorial Conference held August 22–26, 2011 in Budapest. De Gruyter, Berlin/Boston, 2013.
236: J. Pintz. On the density theorem of Halász and Turán. Acta Math. Hungar., 166(1):48–56, 2022.
237: J. Pintz. Density theorems for Riemann’s zeta-function near the line \(\mathrm{Re}\, s = 1\). Acta Arithmetica, 208(1):1–13, 2023.
238: D. Platt and T. Trudgian. The Riemann hypothesis is true up to \(3\cdot 10^{12}\). Bull. Lond. Math. Soc., 53(3):792–797, 2021.
239: G. Pólya. über trigonometrische Integrale mit nur reellen Nullstellen. J. Reine Angew. Math., 158:6–18, 1927.
240: D. H. J. Polymath. Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences, 1(1):12, December 2014.
241: D. H. J. Polymath. Effective approximation of heat flow evolution of the Riemann \(\xi \) function, and a new upper bound for the de Bruijn-Newman constant. Res. Math. Sci., 6(3):Paper No. 31, 67, 2019.
242: O. Ramaré. On schnirelmann’s constant. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Ser. 4, 22(4):645–706, 1995.
243: Olivier Ramaré. An explicit density estimate for dirichlet l-series. Mathematics of Computation, 85(297):325–356, 2016.
244: R. A. Rankin. The difference between consecutive prime numbers. Journal of the London Mathematical Society, s1-13(4):242–247, October 1938.
245: R. A. Rankin. van der Corput’s method and the theory of exponent pairs. The Quarterly Journal of Mathematics, 6(1):147–153, 1955.
246: R. A. Rankin. The difference between consecutive prime numbers V. Proceedings of the Edinburgh Mathematical Society, 13(4):331–332, 1963.
247: H. E. Richert. Verschärfung der Abschätzung beim Dirichletschen Teilerproblem. Mathematische Zeitschrift, 58(1):204–218, December 1953.
248: H. E. Richert. Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen \(\sigma = 1\). Mathematische Annalen, 169(1):97–101, March 1967.
249: O. Robert. An analogue of van der Corput’s \(A^5\)-process for exponential sums. Mathematika, 49(1-2):167–183, 2002.
250: O. Robert. Quelques paires d’exposants par la méthode de Vinogradov. J. Théor. Nombres Bordeaux, 14(1):271–285, 2002.
251: O. Robert. On the fourth derivative test for exponential sums. Forum Mathematicum, 28(2):403–404, 2016.
252: O. Robert. On van der Corput’s \(k\)-th derivative test for exponential sums. Indag. Math. (N.S.), 27(2):559–589, 2016.
253: O. Robert and P. Sargos. Applications des systèmes diophantiens aux Sommes d’exponentielles. PhD thesis, Nancy 1, 2001.
254: O. Robert and P. Sargos. A fourth derivative test for exponential sums. Compositio Mathematica, 130(3):275–292, 2002.
255: B. Rodgers and T. Tao. The de Bruijn–Newman constant is non-negative. Forum Math. Pi, 8:e6, 62, 2020.
256: I. Z. Ruzsa. Essential components. Proceedings of the London Mathematical Society, s3-54(1):38–56, 01 1987.
257: Y. Saouter, X. Gourdon, and P. Demichel. An improved lower bound for the de Bruijn-Newman constant. Math. Comp., 80(276):2281–2287, 2011.
258: P. Sargos. Points Entiers Au Voisinage D’une Courbe, Sommes Trigonométriques Courtes ET Paires D’exposants. Proceedings of the London Mathematical Society, s3-70(2):285–312, 1995.
259: P. Sargos. An analog of van der Corput’s \({A}^4\)-process for exponential sums. Acta Arithmetica, 110(3):219–231, 2003.
260: E. Schmidt. Über die Anzahl der Primzahlen unter gegebener Grenze. Mathematische Annalen, 57(2):195–204, 1903.
261: L. Schnirelmann. Über additive Eigenschaften von Zahlen. Mathematische Annalen, 107(1):649–690, December 1933.
262: A. Schönhage. Eine Bemerkung zur Konstruktion großer Primzahllücken. Archiv der Mathematik, 14(1):29–30, December 1963.
263: A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943.
264: A. Selberg. Contributions to the theory of the Riemann zeta-function. Arch. Math. Naturvid., 48(5):89–155, 1946.
265: A. Selberg. Collected papers. II. Springer Collected Works in Mathematics. Springer, Heidelberg, 2014. Reprint of the 1991 edition [MR1295844], With a foreword by K. Chandrasekharan.
266: W. Sierpiński. O pewnem zagadnieniu z rachunku funkcyj asymptotycznych. Prace Mat. Fiz., 17:77–118, 1906.
267: A. Simonič. Explicit zero density estimate for the Riemann zeta-function near the critical line. J. Math. Anal. Appl., 491(1):124303, 41, 2020.
268: J. Stadlmann. On the mean square gap between primes, 2022.
269: T. Tao. A cheap version of nonstandard analysis. https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/. Accessed: 2024-08-02.
270: T. Tao, T. Trudgian, and A. Yang. New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach, 2025.
271: H. J. J. te Riele. A new lower bound for the de Bruijn-Newman constant. Numer. Math., 58(6):661–667, 1991.
272: E. C. Titchmarsh. A divisor problem. Rend. Circ. Matem. Palermo, 52(1):414–429, 12 1930.
273: E. C. Titchmarsh. On van der Corput’s method and the zeta-function of Riemann (II). The Quarterly Journal of Mathematics, os-2(1):313–320, 01 1931.
274: E. C. Titchmarsh. The Lattice-Points in a Circle. Proceedings of the London Mathematical Society, s2-38(1):96–115, 1935.
275: E. C. Titchmarsh. On divisor problems. The Quarterly Journal of Mathematics, os-9(1):216–220, 1938.
276: E. C. Titchmarsh. On the order of \(\zeta (1/2 + it)\). The Quarterly Journal of Mathematics, os-13(1):11–17, 1942.
277: E. C. Titchmarsh. The theory of the Riemann zeta-function. Oxford Science Publications, Oxford, 1986.
278: K. C. Tong. On divisor problems. J. Chinese Math. Soc., 2:258–266, 1953.
279: T. Trudgian and A. Yang. Toward optimal exponent pairs, 2023.
280: P. Turán. On a density theorem of Yu. V. Linnik. Magyar Tud. Akad. Mat. Kutató Int. Közl., 6:165–179, 1961.
281: P. Turán, G. Halász, and J. Pintz. On a new method of analysis and its applications. Pure and applied mathematics. Wiley-Interscience, New York, 1984.
282: J. G. Van Der Corput. Neue zahlentheoretische Abschätzungen. Mathematische Annalen, 89(3-4):215–254, September 1923.
283: J. H. van Lint and H.-E. Richert. On primes in arithmetic progressions. Acta Arith., 11:209–216, 1965.
284: R. C. Vaughan. On Waring’s problem for smaller exponents. Proceedings of the London Mathematical Society, s3-52(3):445–463, May 1986.
285: I. M. Vinogradov. Eine neue absch ätzung der funktion \(\zeta (1+it)\). Izv. Akad. Nauk SSSR, Ser. Mat., 22:161–164, 1958.
286: G. Voronoi. Sur un problème du calcul des fonctions asymptotiques. Journal für die reine und angewandte Mathematik, 126:241–282, 1903.
287: A. Walfisz. Zur abschätzung von \(\zeta (1/2 + it)\). Nachr. Ges. Wiss. Göttingen, math.-phys, pages 155–158, 1924.
288: A. Walfisz. Über zwei Gitterpunktprobleme. Mathematische Annalen, 95(1):69–83, 1926.
289: A. Walfisz. Teilerprobleme. Mathematische Zeitschrift, 26(1):66–88, December 1927.
290: A. Walfisz. Zur additiven Zahlentheorie. II. Math. Z., 40(1):592–607, 1936.
291: W. Wang. On the least prime in an arithmetic progression. Acta Math. Sinica, 29(6):826–836, 1986.
292: W. Wang. On the least prime in an arithmetic progression. Acta Math. Sinica (N.S.), 7(3):279–289, 1991. A Chinese summary appears in Acta Math. Sinica 35 (1992), no. 4, 575.
293: N. Watt. Exponential sums and the Riemann zeta-function II. Journal of the London Mathematical Society, s2-39(3):385–404, 1989.
294: N. Watt. Short intervals almost all containing primes. Acta Arith., 72(2):131–167, 1995.
295: E. Westzynthius. Über die Verteilung der Zahlen, die zu den \(n\) ersten Primzahlen teilerfremd sind. Commentationes Helsingfors 5, Nr. 25, 37 S. (1931)., 1931.
296: R. E. Wild. On the number of primitive Pythagorean triangles with area less than \(n\). Pacific J. Math., 5:85–91, 1955.
297: D. Wolke. Grosse Differenzen zwischen aufeinanderfolgenden Primzahlen. Math. Ann., 218(3):269–271, 1975.
298: K. C. Wong. Contribution to analytic number theory. PhD thesis, Cardiff, 1996.
299: T. D. Wooley. Large improvements in Waring’s problem. Annals of Mathematics, 135(1):131–164, 1992.
300: T. D. Wooley. On Waring’s problem for intermediate powers. Acta Arithmetica, 176(3):241–247, 2016.
301: P. Xi and J. Zheng. On the Brun–Titchmarsh theorem, 2024.
302: P. Xi and J. Zheng. On the Brun–Titchmarsh theorem. I, 2024.
303: P. Xi and J. Zheng. On the Brun–Titchmarsh theorem. II, 2025.
304: T. Xylouris. Über die Linniksche Konstante. PhD thesis, Universität Heidelberg, 2009.
305: T. Xylouris. On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet \(L\)-functions. Acta Arith., 150(1):65–91, 2011.
306: T. Xylouris. Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression. PhD thesis, Universität Bonn, 2011.
307: T. Xylouris. Linniks Konstante ist kleiner als 5. Chebyshevskii sbornik, 19(3):80–94, 2018.
308: W. L. Yin. Divisor problem for \(d_3(n)\). Science Record, 3:131–134, 1959.
309: W. L. Yin. On divisor problem for \(d_3(n)\). Journal of Peking University (Natural Science Edition), 3:193–196, 1959.
310: W. L. Yin. On divisor problem for \(k=3\). Journal of Sichuan University (Natural Science Edition), 1:31–39, 1964.
311: W. L. Yin and Z. F. Li. Improvement of error term estimation on three dimensional divisor problem. Acta Mathematica Sinica, 24:865–878, 1981.
312: G. Yu. The differences between consecutive primes. Bulletin of the London Mathematical Society, 28(3):242–248, May 1996.
313: M. Y. Yue. A divisor problem. Acta Math. Sinica, 8:496–506, 1958.
314: M. Y. Yue and F. Wu. On the divisor problem for \(d_3(n)\). Sci. Sinica, 11:1055–1060, 1962.
315: Y. Zhang. Bounded gaps between primes. Annals of Mathematics, 179(3):1121–1174, May 2014.
316: L. Zhao. On the Waring–Goldbach problem for fourth and sixth powers. Proc. London Math. Soc., 108(3):1593–1622, 2014.
317: Q. Zheng. A note on three dimensional divisor problem. Journal of Sichuan University (Natural Science Edition), 24(3):376–377, 1988.